Many-body kinetics of dynamic nuclear polarization by the cross effect

A. Karabanov, D. Wisniewski, F. Raimondi, I. Lesanovsky, and W. KockenbergerSchool of Physics and Astronomy, University of Nottingham,

University Park, NG7 2RD, Nottingham, UK andCentre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems,

University of Nottingham, Nottingham NG7 2RD, UK(Dated: March 14, 2018)

Dynamic nuclear polarization (DNP) is an out-of-equilibrium method for generating non-thermalspin polarization which provides large signal enhancements in modern diagnostic methods basedon nuclear magnetic resonance. A particular instance is cross effect DNP, which involves the in-teraction of two coupled electrons with the nuclear spin ensemble. Here we develop a theory forthis important DNP mechanism and show that the non-equilibrium nuclear polarization build-upis effectively driven by three-body incoherent Markovian dissipative processes involving simultane-ous state changes of two electrons and one nucleus. We identify different parameter regimes foreffective polarization transfer and discuss under which conditions the polarization dynamics can besimulated by classical kinetic Monte-Carlo methods. Our theoretical approach allows for the firsttime simulations of the polarization dynamics on an individual spin level for ensembles consistingof hundreds of nuclear spins. The insight obtained by these simulations can be used to find optimalexperimental conditions for cross effect DNP and to design tailored radical systems that provideoptimal DNP efficiency.

Introduction — Spectroscopy and imaging techniquesbased on nuclear magnetic resonance (NMR) are used inmany important applications, ranging from materials sci-ences to biophysics and medical diagnostics. The NMRsignal arises from the Zeeman effect, which at thermalequilibrium gives rise to a weak polarization of the nu-clear spins. The sensitivity of NMR can be significantlyenhanced by dynamic nuclear polarization (DNP), anout-of-equilibrium method that involves the microwavedriven transfer of the much stronger polarization of un-paired electrons to the nuclear spin ensemble via the elec-tron nuclear hyperfine interaction [1–3]. The resultingmany-body dynamics is highly intricate and in particu-lar depends on whether the electrons hosted by param-agnetic centres interact or not. In the case that they donot or only weakly interact, the effective mechanism forpolarization transfer is the solid effect (SE) [4–10], whichcan be understood within the framework of a central spinmodel formed by an isolated electron and its nuclear sur-rounding [11].In this work we focus on the far more involved situationin which coupled electron spins interact collectively withthe nuclear ensemble, thereby creating a polarized out-of-equilibrium state. We consider the particularly relevantscenario of two dipolar coupled unpaired electrons, whichcan be found in biradical molecules [12–14], or when twomonoradicals are in close proximity [15, 16]. Here, acollectively enhanced polarization transfer between elec-trons and the nuclear ensemble can occur via the crosseffect (CE) [17–24]. This DNP mechanism was initallyproposed using phenomenological rate equation modelsfor the bulk electron and nuclear polarization [20, 22, 23].A first attempt to desribe CE DNP quantum mechan-ically for a 3-spin system, albeit neglecting incoherentprocesses, was published by Hu et al. [18]; Hovav et al.were the first to publish a quantum mechanical model

for small spin ensembles (< 12 spins) that also incorpo-rates relaxation processes [17]. Challenging for the simu-lation of such a system are the large differences betweenthe interaction strengths and between the time constantsdescribing the incoherent relaxation processes. Here weshed light on the underlying complex microscopic dynam-ics that is generally governed by an interplay of coherentand incoherent processes and derive efficiency conditionsof CE DNP and its interplay with SE DNP. Moreover,we show under which circumstances the CE DNP non-equilibrium dynamics can be efficiently simulated withclassical kinetic Monte-Carlo methods. This, for the firsttime, enables the investigation of the polarization dy-namics of hundreds of interacting nuclei on an individualspin level. Our study paves the way towards a systematicanalysis, utilization and optimization of realistic many-body CE DNP and may serve as a guidance for the de-sign of optimal DNP regimes and tailor-made polarisingagents (biradicals) [25–31]. We expect our insights tobe also applicable to the non-equilibrium dynamics ofnitrogen-vacancies in diamond [32, 33], which is becom-ing a popular platform for the implementation of quan-tum sensing and diagnostics. Furthermore, our strategycould be applied to study the quantum dynamics of cou-pled quantum dots [34, 35].Model — The model system that we study is schemat-ically shown in FIG. 1(a). It consists of two microwavedriven unpaired electron spins Sj , j = 1, 2, coupled toa large number of nuclear spins Ik, k = 1, 2, . . . , N (as-sumed to be all spin-1/2) at high static field in terms ofa Markovian Lindblad master equation for the densitymatrix ρ in the rotating wave approximation:

ρ = −i[H, ρ] +Dρ.

The Hamiltonian H = HZ + HMW + Hint describesthe Zeeman splitting HZ = ωIIz +

∑j ∆jSjz (Iz =

2

FIG. 1. (a) Schematic representation of the model system (see text for the definition of symbols). (b) Resonance structure ofthe Zeeman eigenstates and schematics of the basic spin processes. (c) Fast and slow stages of the polarization dynamics ofthe n − e1 − e2 model simulated with the full master equation with p = 0.034 (ωS = 100 GHz, T = 70 K), ωI = 145 MHz,ω1 = D = 0.5 MHz, B = 0.1 MHz, T1e = 3 ms, T2e = 50 µs, T1n = 100 s, T2n = 1 ms. Here the efficiency condition (3) isfulfilled with η1 ∼ 1 leading to strongly coherent Rabi oscillations of the first electron ( S1, blue curve) in contrast with theincoherent evolution of the nucleus ( I1, yellow curve). The dashed lines show a simulation with p = 0.98 (T = 1 K), ω1 = 10kHz, D = 100 kHz, T1e = 1 s (all other parameters identical to the coherent case), corresponding to η1 ∼ 100. In this case onlyincoherent dynamics is observed. All polarization levels are normalized to the thermal electron polarization p.

∑k Ikz, ∆j are the offsets of the electron Larmor fre-

quencies from the microwave frequency and ωI is the nu-clear Larmor frequency), microwave irradiation HMW =ω1

∑j Sjx (with the strength of the microwave field

ω1) and electron, nuclear and electron-nuclear spin in-teractions Hint = HSS + HII + HSI . The electronand nuclear interactions are represented by dipole-dipolesecular terms: HSS = D(3S1zS2z − S1 · S2), withthe electron-electron coupling strength D and HII =∑

k<k′ dkk′(3IkzIk′z − Ik · Ik′) with the nuclear inter-action strengths dkk′ . The electron-nuclear interac-tions are described by secular and semi-secular parts:HSI =

∑k,j(AjkIkz + BjkIk+/2 + B∗jkIk−/2)Sjz with

the respective interaction strengths Ajk and Bjk. Relax-ation is represented by a single-spin Lindblad dissipatorD = DS + DI , accounting for electron and nuclear con-tributions,

DS =∑

j [Γ1+L(Sj+) + Γ1−L(Sj−) + Γ2L(Sjz)],

DI =∑

k[γ1(L(Ik+) + L(Ik−)) + γ2L(Ikz)],

L(X)ρ ≡ XρX† − (X†Xρ+ ρX†X)/2,

(1)

with Γ1± = (1∓ p)R1/2, Γ2 = 2R2, γ1 = r1/2, γ2 = 2r2.Here R1,2, r1,2 are the electron and nuclear longitudi-nal and transversal relaxation rates, respectively. Theparameter p = tanh(~ωS/2kBT ) ∈ (0, 1) is the electronthermal polarization that depends on the average elec-tron Larmor frequency ωS and temperature T .Conditions for efficient CE DNP — A fundamentalprerequisite for CE DNP is that the Larmor frequenciesof the two electrons are separated by the nuclear Lar-mor frequency, ∆2 −∆1 ∼ ωI . Under this condition thefirst electron is saturated more efficiently than the secondand the arising polarization difference between the two istransferred to a coupled nuclear spin by a three-spin pro-cess. In the following we assume that the offsets are cho-sen such that the matching condition ∆1 = ∆2 − ωI = 0is fulfilled, i.e. the microwave field is applied on reso-

nance with the Larmor frequency of the first electron. Ina typical high-field DNP experiment |D|, |Bjk| � |ωI |and the electron saturation predominately depends onthe microwave term and Zeeman orders of the spin cou-plings. The projections of the Hamiltonian to the sub-spaces of the electron spins become Hj = ω1Sjx +∆jSjz,j = 1, 2. Here ∆j are the effective (operator-valued)electron spin offsets ∆1 = 2DS2z +

∑k A1kIkz, ∆2 =

ωI + 2DS1z +∑

k A2kIkz which take on discrete valuesdepending on the up and down orientations of spins. Toidentify conditions for which the polarization differencebetween the two electrons is maximal we consider first asingle electron driven by a microwave field at an offset∆. For its relative polarization components X,Y and Zdefined by ρ = 1/2− p (XSx + Y Sy + ZSz) , we can usethe Bloch equations (for R2 � R1)

X = −∆Y −R2X, Y = ∆X − ω1Z −R2Y,

Z = ω1Y +R1(1− Z),(2)

with unique stationary steady-state solution

X =∆Z

ω1η′, Y = −R2Z

ω1η′, Z =

(1 +

η

η′

)−1,

η′ =R2

2 + ∆2

ω21

, η =R2

R1,

to describe the saturation of the spin. The ratio ε = η/η′

is a direct indicator of the saturation of the spin: we havea full saturation Z ∼ 0 for ε� 1 and no saturation Z ∼ 1if ε� 1. Hence, extending the notations ∆, η′, Z to twoelectrons ∆j , ηj , pj/p, we can find conditions (assumedto be fulfilled for all discrete values of ∆j) for the fullsaturation of the first electron and unchanged polarisa-tion of the second electron, i.e., the maximal polarisationdifference between the electrons. For

η1 � η � η2, ηj = (R22 + ∆2

j )/ω21 , η = R2/R1, (3)

3

FIG. 2. Steady-state nuclear polarization enhancement (a),build-up time (b) and error between the full master equationand the Zeeman projection (c) as functions of the electron-electron coupling D for different values of the microwave fieldstrength ω1. In (a,b) calculations are made for the 3-spin e1−e2 − n model. Calculations in (c) were made with the 5-spinpentagon configuration of FIG. 1(a) with p = 0.984 (ωS = 100GHz, T = 1 K), B11 = B22 = 0.1 MHz, d13 = d23 = 10Hz (other interaction constants are set to zero), R1 = 1s−1,R2 = 105 s−1 and other parameters as in FIG. 1(c). (d)Individual polarization build-up of 118 randomly distributedprotons obtained using the Lindblad master equation (4) andkMC (averaged over 2000 trajectories). The two electronsare shown as red dots (larger diameter). The polarization ofthe nuclei relative to the thermal polarization of the electronis shown by a grey-blue colour scale and dot size (see scaleat the right) after a short-term (10 s) and long-term (220s) evolution. (e) Total nuclear polarization build-up for themodel described in (d). The effect of spin diffusion can clearlybe seen by comparing the build-up in a system with nucleardipole interaction (solid) to a system in which the dipolarinteraction is set to zero (dashed). The enhancement ε andthe buildup time τDNP, τ in (b,d) are obtained by fitting amonoexponential function for the total nuclear polarizationpI = ε(1 − exp [−t/τ ]).

the first electron is almost fully saturated, p1 � p, whilethe second electron is approximately thermal, p2 ∼ p,and a maximal polarization difference between the elec-trons is generated. Thus, condition (3) defines a param-eter regime in which CE DNP is efficient.Polarization of single nucleus — In order to get afirst idea of the timescales and the role of coherent andincoherent processes in CE DNP, we consider the sim-plest case of two electrons and a single nucleus coupledonly to the first electron via the semi-secular interactionof strength B. For simplicity we ignore the secular part

of the electron-nuclear interaction A (we refer to this asthe n− e1 − e2 model, see [17] for a detailed discussion).At the resonance ∆1 = ∆2 − ωI = 0, the 8-levelZeeman Hamiltonian HZ = ωI(S2z + Iz) in the mi-crowave rotating frame has three sets of degenerate eigen-states with energies 0, ±ωI [see FIG. 1(b)]. The den-sity matrix at thermal equilibrium is well approximatedby ρth =

∏j(1 − 2pSjz)/8, with the thermal polariza-

tions of the electrons and nucleus being p1 = p2 = p,pI = 0, respectively. Proceeding to the full masterequation, the non-Zeeman terms of the Hamiltonian con-nect the Zeeman eigenstates and cause a population ex-change between them. The microwave term ω1S1x (re-sponsible for the saturation of the first electron) con-nects states with the same energy. The microwave termω1S2x, the flip-flop part of the electron-electron coupling−D(S1+S2− + S1−S2+)/2 and the semi-secular part ofthe electron-nuclear interaction (BjkIk+ +B∗jkIk−)Sjz/2

(mediating the nuclear polarization build-up) connectstates with different energies. Due to |D|, |B| � |ωI |,the exchange within the degenerate manifolds occurs ona much faster time-scale than the exchange between thedegenerate manifolds. Therefore, the polarization dy-namics starting from the thermal state can be dividedinto two stages: the fast stage of the first electron sat-uration and the slow stage of the nuclear polarizationbuild-up.In the fast stage, the polarization of the first electron be-comes zero p1 = 0 while polarization levels of the secondelectron and nucleus remain unchanged p2 = p, pI = 0.Inspecting again the Bloch equation for a single elec-tron (2), two different cases for the transient dynamicsoccurring until the electron is saturated can be distin-guished. The dynamics described by the Bloch equa-tions (2) can be separated into the transverse dynam-ics that arises from spin precession and T2-induced re-laxation and the longitudinal dynamics caused by T1-induced relaxation. These two dynamics are correlatedby the microwave term with the microwave field strengthω1 that couples the two equations for Y and Z. IfR2

2 + ∆2 � ω21 , R

21, the correlation between these two

dynamics becomes negligible and the transverse dynam-ics can be adiabatically eliminated and an expression forthe longitudinal dynamics can be found by substituingthe steady state solution for Y into the Bloch equationfor Z, Z = R1 − (R1 + R2/η

′)Z. Since R2 � R1 theadiabatic elimination can be carried out if η′ � 1. Ap-plying this discussion to the first electron, we see thatonly for small or intermediate values of η1, the fast sat-uration of the first electron is accompanied by coherentRabi oscillations, arising from a strong correlation be-tween the observable longitudinal polarization dynamics〈S1z〉 and the dynamics in the transversal plane 〈S1x,y〉[see solid blue curve in FIG. 1(c) corresponding to S1].On the contrary, for η1 � 1 the microwave irradiationdoes not cause Rabi oscillations and the Bloch vectorof the first electron remains parallel to the static fieldwhile displaying an incoherent longitudinal polarization

4

dynamics 〈S1z〉 [dashed lines in FIG. 1(c)]. We will re-turn to this important distinction further below. Duringthe slow stage, the electron-electron flip-flops are energet-ically matched with the nuclear flips, and effective tripleelectron-electron-nuclear flips occur that tend to equili-brate the populations of the states |↓↑↓〉, |↑↓↑〉 [17, 19–22]. As a consequence, the thermal polarization of thefirst electron, which is saturated during the fast stage,is fully transferred to the coupled nucleus [solid yellowcurve in FIG.1(c)].The efficiency of CE DNP compared to SE DNP is better,particularly at low temperatures and weak microwavefields, because the nuclear polarization build-up time ismuch shorter and the nuclear polarization enhancementis much higher. To illustrate this, we consider again thesimplest three-spin case but now with a nucleus coupledonly to the second electron (the e1−e2−n model). Unlikein the n − e1 − e2 model, where CE DNP is the exclu-sive polarization transfer mechanism, in the e1 − e2 − nmodel both CE and SE DNP mechanisms can simulta-neously participate in the dynamics [17]. In FIG. 2(a,b),typical plots of the steady-state nuclear polarization en-hancement and build-up time are shown as a functionof the electron-electron coupling strength D for differentmicrowave field strengths ω1. For small ω1, the steady-state polarization is an order of magnitude higher andthe build-up an order of magnitude faster for the CE case(D ≈ 100 kHz) compared to the pure SE case (D = 0,note that this is identical to the steady-state polarizationfor D = 1kHz shown in Fig. 2(c))). These differences de-crease with increasing ω1.Representation by incoherent Markovian dynam-ics — In order to gain insight to the realistic many-body CE DNP dynamics with, e.g., the goal to optimizethe physical system parameters and spin geometries, oneneeds to consider large nuclear ensembles. In previouswork we have shown for the case of SE DNP that it ispossible to reduce the quantum master equation dynam-ics to a set of kinetically-constrained rate equations thatcan be efficiently simulated [11].Such a strategy is also possible for CE DNP, providedthat the saturation of the first electron dominating thefast stage dynamics is incoherent. Generalizing the pro-jective adiabatic elimination technique detailed in [11],we obtain a Lindbladian master equation for the den-sity operator ρZ , which depends on single, two-body andthree-body spin flip processes:

ρZ = DρZ , D = D1 +D2 +D3. (4)

Here the single-spin dissipator

D1 =∑2

j=1

[ΓSj+L(Sj+) + ΓS

j−L(Sj−)]

+∑Nk=1 ΓI

kL(Ik+ + Ik−)

and the rates ΓSj±, ΓI

k arise from spin relaxation, mi-crowave saturation of the electrons and the effective semi-secular saturation of the nuclei. In the double-spin dissi-

pator

D2 = ΓSSL(K +K†) +∑N

k=1 ΓSIk L(Yk + Y †k )+∑

k<k′ ΓIIkk′L(Xkk′ +X†kk′),

K = S1+S2−, Xkk′ = Ik+Ik′−, Yk = S2−Ik+,

the rates ΓSS , ΓIIkk′ characterize the inter-electron and

internuclear flip-flops caused by the dipolar interactionswithin the electron and nuclear ensembles and the ratesΓSIk describe the effective electron-nuclear flip-flops due

to the SE resonance of the second electron and the nuclei.The triple-spin dissipator

D3 =∑N

k=1 ΓSSIk L(Zk + Z†k), Zk = S1+S2−Ik+,

and the rates ΓSSIk represent the effective CE electron-

electron-nuclear flips. [For full details and explicit ex-pressions for the Lindbladian rates see SM A].In the following we analyze in more detail the conditionsunder which this effectively classical description of CEDNP is applicable. As described in the previous sec-tion and illustrated in FIG. 1(c), to exclude the stronglycoherent saturation dynamics of the first electron, we re-quire η1 � 1, which guarantees the absence of Rabi os-cillations. Combined with condition (3) this leads to atriple inequality that defines a balance between the sys-tem parameters for which CE DNP is efficient and thesaturation of the first electron is incoherent. In partic-ular, it implies η = R2/R1 � 1, which is realized ina low-temperature regime. Since the square of the mi-crowave field strength ω1 is in the denominator of η1,condition η1 � 1 also implies a weak microwave field.Such conditions are typically fulfilled in cryogenic tem-perature DNP where microwave sources of only a few100s mW are used. To illustrate the importance of thiscondition, we considered a 5-spin system consisting oftwo electrons S1,2 and three nuclei I1−3 arranged in apentagon configuration that represents two “core” nucleiin close vicinity to the electrons and a “bulk” nucleus re-mote from the electrons, c.f. FIG. 1(a), with a symmetricset of interaction strengths. For this small representativesystem, the exact numerical solution of the full masterequation can be compared with the Zeeman projection.The result is shown in FIG. 2(c) where we plotted themaximal error ∆pI/p, ∆pI ≡ maxt,k

∣∣pFEI,k(t) − pZPI,k(t)∣∣

(normalized to the thermal electron polarization), overthe nuclear polarization build-up between the full mas-ter equation and the Zeeman projection as a function ofthe electron-electron coupling D for different values ofthe microwave field strength ω1. It is evident that theerror is small for large η1 (calculated for values of D atthe error peaks) and increases with decreasing η1 (stillremaining < 10% for ω1 < 30 kHz). Note that the max-imal build-up error is obtained at the steady-state, andin the case D = 0 the dynamics of the first electron isdecoupled and does not influence the error.In contrast to SE DNP, there are electron-electron and

5

three-body electron-electron-nuclear jumps in the effec-tive Lindbladian (4) of CE DNP. The correspondingrates are given by ΓSS = D2R2/2ω

2I , ΓSSI

k = D2|B1k −B2k|2τ3k/ω2

I , where the (operator valued) magnitudes τ3khave dimension of time and depend on the spin transver-sal relaxation rates r2, R2 and secular electron-nuclearinteraction strengths Ajk (see SM A for the full expres-sions). For a good approximation of the polarization dy-namics, the condition ΓSS , ΓSSI

k � R2 must be fulfilledthat guarantees that the relevant spin jumps are repre-sented by incoherent Markovian processes (see SM A fordetails). The other single- and double-spin effective ratesin D1,2 are similar to those described in the SE case [11].Comparing the CE triple-spin rates ΓSSI

k with the SEdouble-spin rates ΓSI

k (given in SM A), we see that forD2 � ω2

1 the CE dominates over the SE while for D ∼ ω1

the CE and SE mechanisms (indicated by the blue andred arrows in FIG. 1(b)) equally influence the dynamics(see FIG. 2(a,b) as an illustration). Note that condition(3) is violated for large values of D, making the DNPprocess inefficient.Where the condition η1 � 1 is not fulfilled, the coherentoscillations in Fig. 1(c), which are not captured by therate equations (4), become important. We conjecture,that in this case it is possible to use the state after thefast coherent oscillation as an initial state and an alter-native incoherent Lindblad equation similar to Eq. (4)can be found to describe the evolution of the system.

Large-scale simulations — To simulate polarizationdynamics of large-scale nuclear ensembles, a kineticMonte Carlo (kMC) scheme can be applied to the in-coherent Lindblad master equation (4) similar to thatdescribed for SE DNP [11] (see also SM for more details).One such simulation is represented in FIG. 2(d,e), show-ing individual polarization build-up for 118 protons 1Hin a random spatial distribution coupled to two electrons.The ideal condition of the electron frequency offsets wasused and the parameters for the spin ensemble were cho-sen to fulfil conditions (3) and η1 � 1. In FIG. 2(e)the two cases of interacting (d 6= 0) and non-interacting(d = 0) protons are compared to demonstrate the crucial

role of nuclear spin diffusion in the build-up of spin po-larization within the nuclear ensemble. The distributionof spin polarization by nuclear spin diffusion simulated inFIG. 2(d,e) is a collective many-body effect of the spinsystem. Particularly in the case of many polarizationsources a large number of nuclei must be considered toobtain a meaningful representation and to analyze condi-tions under which different DNP mechanisms operate inparallel [15]. This was impossible with previous theoreti-cal approaches. Our model allows the CE DNP efficiencyto be calculated taking into account the spin geometryof radical molecules embedded in a glass forming matrix.Therefore it provides an important step towards a morerational and less empirical design of radical compoundsthat have optimized DNP efficiencies.In the next step the polarization dynamics predicted byour model needs to be compared to experimental data.The experimentally measured changes of the nuclear po-larization arise only from the bulk nuclei (with nucleiclose to the electron not contributing to the signal be-cause of their large hyperfine frequency shift). The amor-phous structure of the sample makes it necessary to cal-culate a powder average by changing the angle that theaxis between the two electrons forms with the direction ofthe applied static magnetic field. For these calculationsit is crucial that the polarization trajectory for a singleorientation can be obtained in a fast and efficient wayto avoid lengthy simulation times. Our model providesboth the possibility to include a large number of nuclearspins in the calculations while still keeping the computa-tional time in a reasonable regime (see SM B). A detailedvalidation study will require comparison between the pre-diction by our model and the experimental polarizationdynamics depending on both parameters such as the rad-ical and the nuclear spin concentration or the microwavefield strength.Acknowledgments — The research leading to these re-sults has received funding from the European ResearchCouncil under the European Union’s Seventh FrameworkProgramme (FP/2007-2013) / ERC Grant AgreementNo. 335266 (ESCQUMA) and the EPSRC Grant No.EP/N03404X/1.

[1] A. W. Overhauser, Phys. Rev. 92, 411 (1953).[2] A. Abragam and M. Goldman, Nuclear Magnetism: Or-

der and Disorder (Oxford Clarendon Press, 1982).[3] W. T. Wenckebach, Essentials of Dynamic Nuclear

Polarization (The Netherlands Spindrift Publications,2016).

[4] C. Jeffries, Phys. Rev. 106, 164 (1957).[5] A. Abragam and W. G. Proctor, C.R Hebd. Seances.

Acad. Sci. 246, 2253 (1958).[6] T. Schmugge and C. Jeffries, Phys. Rev. 138, A1785

(1965).[7] C. Jeffries, Dynamic Nuclear Orientation (New York

John Wiley, 1963).[8] M. Borghini and A. Abragam, C.R Hebd. Seances. Acad.

Sci. 203, 1803 (1959).[9] A. Abragam and M. Borghini (North-Holland, Amster-

dam, 1964).[10] Y. Hovav, A. Feintuch, and S. Vega, J. Magn. Reson.

207, 176 (2010).[11] A. Karabanov, D. Wisniewski, I. Lesanovsky, and

W.Kockenberger, Phys. Rev. Lett. 115, 020404 (2015).[12] C. Song, K. Hu, C. Joo, T. Swager, and R. Griffin, J.

Am. Chem. Soc. 128, 11385 (2006).[13] Y. Matsuki, T. Maly, O. Ouari, H. Karoui, F. L.

Moigne, E. Rizzato, S. Lyubenova, J. Herzfeld, T. Pris-ner, P.Tordo, and R. Griffin, Angew. Chem. Intl. Ed.48, 4996 (2009).

[14] K. Hu, H. Yu, T. Swager, and R. Griffin, J. Am. Chem.

6

Soc. 126, 10844 (2004).[15] D. Shimon, Y. Hovav, A. Feintuch, D. Goldfarb, and

S. Vega, Phys. Chem. Chem. Phys. 14, 5729 (2012).[16] Y. Hovav, O. Levinkron, A. Feintuch, and S. Vega, Appl.

Magn. Reson. 43, 21 (2012).[17] Y. Hovav, A. Feintuch, and S. Vega, J. Magn. Reson.

214, 29 (2012).[18] K.-N. Hu, G. Debelouchina, A. Smith, and R. Griffin, J.

Chem Phys. 134, 125105 (2011).[19] A. Kessenikh, V. Lushchikov, A. Manenkov, and

Y. Taran, Sov. Phys. Solid State 5, 321 (1963).[20] A. Kessenikh, A. Manenkov, and G. Pyatnitskii, Sov.

Phys. Solid State 6, 641 (1964).[21] C. Hwang and D. Hill, Phys. Rev. Lett. 18, 110 (1967).[22] C. Hwang and D. Hill, Phys. Rev. Lett. 19, 1011 (1967).[23] D. Wollan, Phys. Rev. B 13, 3671 (1976).[24] D. Wollan, Phys. Rev. B 13, 3686 (1976).[25] K.-N. Hu, H. Yu, T. M. Swager, and R. G. Griffin, J.

Am Chem. Soc. 126, 10844 (2004).[26] C. Song, K.-N. Hu, C.-G. Joo, T. M. Swager, and R. G.

Griffin, J. Am Chem. Soc. 128, 11385 (2006).[27] C. Ysacco, E. Rizzato, M.-A. Virolleaud, H. Karoui,

A. Rockenbauer, F. L. Moigne, D. Siri, O. Ouari, R. G.

Griffin, and P. Tordo, Phys. Che. Chem. Phys 12, 5841(2010).

[28] G. Mathies, M. Caporini, V. k. Michaelis, Y. Liu, K.-N.Hu, D. Mance, J. L. Zweier, M. Rosay, M. Baldus, andR. G. Griffin, Angew. Chem. Int. Ed. 54, 11770 (2015).

[29] D. J. Kubicki, G. Casano, M. Schwarzwalder, S. Abel,C. Sauvee, K. Ganesan, M. Yulikov, A. J. Rossini,G. Jeschke, C. Coperet, A. Lesage, P. Tordo, O. Ouari,and L. Emsley, Chem. Sci 7, 550 (2016).

[30] D. J. Kubicki and L. Emsley, Chimia 71, 190 (2017).[31] F. Perras, A. Sadow, and M. Pruski, ChemPhysChem ,

10.1002/cphc.201700299 (2017).[32] Z.-Y. Wang, J. F. Haase, J. Casanova, and M. Plenio,

Phys. Rev. B 93, 174104 (2016).[33] P. London, J. Scheuer, J. Cai, I. Schwarz, A. Retzker,

M. Plenio, M. Katagiri, T. Teraji, S. Koizumi, J. Isoya,R. Fischer, L. McGuinness, B. Naydenov, and F. Jelezko,Phys. Rev. Lett. 111, 067601 (2013).

[34] J. Petta, A. Johnson, J. Taylor, E. Laird, A. Yacoby,M. Lukin, C. Marcus, M. Hanson, and A. Gossard, Sci-ence 309, 2180 (2005).

[35] I. Bragar and L. Cywinski, Phys. Rev. B 91, 155310(2015).